Review AP Calculus AB/BC Unit 6 to build fluency with integration, the Fundamental Theorem of Calculus, and every major antidifferentiation technique. This unit carries 17-20% of the exam weight and connects directly to Units 7 and 8, so strong command here pays off across the entire course.
Use the topic guides, key terms, and practice questions available for this unit to work through all 14 topics before your exam.
Unit 6 is the integration half of AP Calculus. Where differentiation finds an instantaneous rate of change, integration finds the total accumulation of change over an interval. The unit moves from geometric intuition through formal notation to a complete set of antidifferentiation techniques.
Topics 6.1-6.3 build the concept: area under a rate-of-change graph equals accumulated change, Riemann sums approximate that area using rectangles or trapezoids, and the definite integral is the limit of those sums as subinterval widths approach zero.
Topics 6.4-6.7 formalize the connection between differentiation and integration. Part 1 says d/dx of the integral from a to x of f(t) dt equals f(x). Part 2 says the definite integral from a to b of f(x) dx equals F(b) minus F(a). Together they make exact evaluation possible.
Topics 6.8-6.14 build the toolkit: basic rules and notation, u-substitution, long division, completing the square, and (BC only) integration by parts, linear partial fractions, improper integrals, and technique selection.
The central idea of Unit 6 is that integration and differentiation are inverse processes. Every technique in the unit serves one goal: finding the exact accumulated change over an interval. The Fundamental Theorem of Calculus is the bridge that makes this possible, and every topic from 6.4 onward depends on it.
Area between a rate-of-change graph and the x-axis equals accumulated change. Positive rate means positive accumulation; negative rate means negative accumulation. Simple shapes can be evaluated with geometry. Units equal rate unit times input unit.
Left, right, midpoint, and trapezoidal sums approximate a definite integral using rectangles or trapezoids. Over- and underestimate behavior depends on whether the function is increasing, decreasing, concave up, or concave down. Works with graphs, tables, equations, or verbal descriptions.
A Riemann sum in sigma notation sums f(x_i*) times delta x_i over all subintervals. As the maximum subinterval width approaches zero, this limit equals the definite integral from a to b of f(x) dx. You should be able to translate between the limit of a sum and integral notation.
FTC Part 1: d/dx of the integral from a to x of f(t) dt = f(x). Accumulation functions g(x) = integral from a to x of f(t) dt are differentiable, with g(a) = 0. When the upper limit is h(x), apply the chain rule to get f(h(x)) times h'(x).
Read g(x) = integral from a to x of f(t) dt from the graph of f. g increases where f > 0, decreases where f < 0, has extrema where f changes sign, and changes concavity where f changes from increasing to decreasing.
Key properties: reversing limits changes sign, additivity over adjacent intervals, constant multiples and sums distribute. Geometric evaluation works when the graph forms triangles, rectangles, or semicircles. Integrals extend to functions with removable or jump discontinuities.
FTC Part 2: the integral from a to b of f(x) dx = F(b) minus F(a) for any antiderivative F of f. Requires f to be continuous on [a, b]. The constant of integration cancels in the subtraction.
Indefinite integral notation: integral of f(x) dx = F(x) + C. Core antiderivatives include the reverse power rule, e^x, 1/x, sin x, cos x, sec^2 x, 1/(1+x^2), and 1/sqrt(1-x^2). Differentiation rules run in reverse.
U-substitution reverses the chain rule. Set u = inner function, replace g'(x) dx with du, integrate in u, back-substitute. For definite integrals, convert the limits to u-values and skip back-substitution.
Use polynomial long division when the numerator degree is greater than or equal to the denominator degree. Complete the square on irreducible quadratic denominators to reach arctan or arcsin antiderivative forms.
Formula: integral of u dv = uv minus the integral of v du. Use LIATE to choose u. Repeat for higher-degree polynomial factors; use the cyclic method when integration by parts returns the original integral.
Decompose a proper rational function with distinct linear factors into A/(ax + b) terms. Solve for constants by substituting roots. Each term integrates to a natural log. Apply long division first if the function is improper.
Replace an infinite bound or a point of unboundedness with a limit variable. Integrate normally, then evaluate the limit. Finite limit means convergence; infinite or nonexistent limit means divergence. Split at interior asymptotes.
Identify integrand structure before computing. Match the structure to the correct technique: basic rules, u-substitution, long division, completing the square, or (BC) integration by parts, partial fractions, or improper integral limits.
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Review Integrating Using Integration by Parts (BC Only) with attention to how the concept appears in AP-style source and evidence questions.
Review Integrating Functions Using Long Division and Completing the Square with attention to how the concept appears in AP-style source and evidence questions.
Review Selecting Techniques for Antidifferentiation with attention to how the concept appears in AP-style source and evidence questions.
Review Riemann Sums, Summation Notation, and Definite Integral Notation with attention to how the concept appears in AP-style source and evidence questions.
The area between a rate-of-change function and the x-axis represents accumulated change. When the rate is positive, accumulation is positive; when negative, accumulation is negative. For simple shapes (rectangles, triangles, semicircles), you can compute the area geometrically without an antiderivative. Units for accumulated change equal the rate unit multiplied by the input unit (for example, miles per hour times hours equals miles).
When geometry is not enough, Riemann sums approximate the definite integral by summing products of function values and subinterval widths. Left, right, midpoint, and trapezoidal sums each have predictable over- or underestimate behavior based on whether the function is increasing, decreasing, concave up, or concave down. As the maximum subinterval width approaches zero, the Riemann sum limit equals the definite integral, written as the integral from a to b of f(x) dx.
| Method | Height used | Overestimates when | Underestimates when |
|---|---|---|---|
| Left sum | Left endpoint | f is decreasing | f is increasing |
| Right sum | Right endpoint | f is increasing | f is decreasing |
| Midpoint sum | Midpoint of subinterval | f is concave down | f is concave up |
| Trapezoidal sum | Average of endpoints | f is concave up | f is concave down |
An accumulation function g(x) = integral from a to x of f(t) dt defines a new function whose output is the net signed area under f from a to x. By FTC Part 1, g'(x) = f(x). This means you read the behavior of g directly from the graph of f: g increases where f is positive, decreases where f is negative, has a local extremum where f changes sign, and changes concavity where f changes from increasing to decreasing. When the upper limit is a composite function h(x), apply the chain rule: d/dx of the integral from a to h(x) of f(t) dt equals f(h(x)) times h'(x).
Definite integral properties let you manipulate and evaluate integrals algebraically. Reversing limits changes the sign; splitting an interval at an interior point uses additivity; constant multiples and sums distribute across the integral sign. FTC Part 2 gives the evaluation formula: the integral from a to b of f(x) dx = F(b) minus F(a), where F is any antiderivative of f. The constant of integration cancels in the subtraction, so you can use any convenient antiderivative.
Every differentiation rule has an integration counterpart. The reverse power rule gives the integral of x^n as x^(n+1)/(n+1) + C for n not equal to -1. Key antiderivatives to memorize include 1/x to ln|x| + C, e^x to e^x + C, sin x to -cos x + C, cos x to sin x + C, sec^2 x to tan x + C, 1/(1+x^2) to arctan x + C, and 1/sqrt(1-x^2) to arcsin x + C. U-substitution reverses the chain rule: set u = g(x), compute du = g'(x) dx, rewrite the integral in terms of u, integrate, then substitute back. For definite integrals, change the limits of integration to u-values instead of substituting back.
When the integrand is a rational function with numerator degree greater than or equal to denominator degree, use polynomial long division to rewrite it as a polynomial plus a proper fraction. When the denominator is an irreducible quadratic, complete the square to convert it to the form (x + a)^2 + b^2, which leads to an arctan antiderivative using the formula: integral of 1/(x^2 + a^2) dx = (1/a) arctan(x/a) + C. Both techniques rearrange the integrand into a form you already know how to integrate.
Integration by parts applies when the integrand is a product of two functions and u-substitution does not work. The formula is: integral of u dv = uv minus the integral of v du. Choose u using the LIATE priority order (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) so that the new integral is simpler. Some integrals require repeated application; cyclic cases (such as e^x sin x) are solved by setting the repeated integral equal to a variable and solving algebraically.
Partial fraction decomposition rewrites a proper rational function whose denominator factors into distinct linear factors as a sum of simpler fractions, each of the form A/(ax + b). Each term integrates to a natural log expression. If the rational function is improper, apply polynomial long division first. Solve for the constants A, B, etc. by substituting the roots of each linear factor into the cleared equation or by matching coefficients.
An improper integral has an infinite limit of integration or an integrand that is unbounded on the interval. Evaluate by replacing the problematic bound with a limit variable, integrating normally, then taking the limit. If the limit is a finite number, the integral converges; if the limit is infinite or does not exist, the integral diverges. When the integrand has a vertical asymptote inside the interval, split the integral at the asymptote and evaluate each piece separately.
Topic 6.14 is about recognition, not a new technique. Given any integrand, identify its structure and match it to the correct method: reverse power rule or known antiderivative for simple forms, u-substitution when an inner function and its derivative both appear, long division when the rational function is improper, completing the square for irreducible quadratic denominators, and (BC) integration by parts for products or isolated logarithms and inverse trig functions, partial fractions for factorable rational denominators, and limits for improper integrals.
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| Term | Definition |
|---|---|
| Antiderivative | A function F whose derivative equals f. Finding antiderivatives is the core task of Unit 6; every integration technique produces an antiderivative. |
| Indefinite Integral | Written as the integral of f(x) dx = F(x) + C; represents the entire family of antiderivatives of f, differing only by the constant C. |
| Constant of Integration | The arbitrary constant C added to every indefinite integral. It cancels when evaluating a definite integral using F(b) minus F(a). |
| Riemann Sum | An approximation of a definite integral computed by summing products of function values and subinterval widths over a partition of [a, b]. |
| Left Riemann Sum | A Riemann sum using the left endpoint of each subinterval as the function value; overestimates when f is decreasing, underestimates when increasing. |
| Right Riemann Sum | A Riemann sum using the right endpoint of each subinterval; overestimates when f is increasing, underestimates when decreasing. |
| trapezoidal sum | Approximates a definite integral by averaging the left and right endpoint values on each subinterval; overestimates when f is concave up. |
| Area under the curve | The signed area between a function and the x-axis over an interval; equals the definite integral and represents accumulated change when f is a rate function. |
| Limits of Integration | The values a and b in the integral from a to b of f(x) dx that specify the interval over which accumulation is measured. |
| U-Substitution | An antidifferentiation technique that reverses the chain rule by substituting u = g(x) and du = g'(x) dx to simplify the integrand. |
| Completing the Square | Rewriting a quadratic expression ax^2 + bx + c in the form a(x + h)^2 + k to convert a denominator into a form that yields an arctan antiderivative. |
| Improper Integral | An integral with an infinite limit of integration or an unbounded integrand; evaluated as the limit of a definite integral and classified as convergent or divergent. |
| Rational Functions | Functions expressed as a ratio of two polynomials; integrated using long division (if improper), completing the square, or partial fractions depending on the denominator structure. |
| Rate of Change | The function f whose graph is used to measure accumulation; the area under a rate-of-change graph over an interval gives the net change in the original quantity. |
When applying u-substitution to a definite integral, the limits of integration are x-values. You must substitute them into u = g(x) to get u-values before evaluating. Evaluating F(u) at the original x-limits gives the wrong answer.
Over- and underestimate depend on both monotonicity and concavity depending on the method. Left and right sums depend on whether f is increasing or decreasing. Trapezoidal sums depend on concavity. Midpoint sums depend on concavity in the opposite direction from trapezoidal.
The integral from b to a of f(x) dx equals the negative of the integral from a to b of f(x) dx. This sign flip is easy to miss when combining integrals over adjacent intervals or when applying FTC Part 2 with a lower variable limit.
When the upper limit is h(x) rather than x, d/dx of the integral from a to h(x) of f(t) dt equals f(h(x)) times h'(x). Forgetting to multiply by h'(x) is one of the most common errors on accumulation function problems.
Every indefinite integral requires plus C. Omitting it is incorrect notation and loses points on free-response questions. The constant cancels in definite integral evaluation, but it must appear in all indefinite integral answers.
AP Calculus free-response questions frequently present a rate function (velocity, flow rate, population growth) and ask you to compute a definite integral, interpret its meaning with correct units, or use FTC Part 2 to find a total accumulated value. Expect to apply both geometric evaluation from a graph and analytical evaluation using an antiderivative.
Multiple-choice and free-response items regularly define a function as g(x) = integral from a to x of f(t) dt and ask for g'(x), g''(x), local extrema of g, or intervals where g is increasing or concave up. These require FTC Part 1, the chain rule for composite upper limits, and the ability to read f's graph to describe g's behavior without finding a formula.
Both AB and BC exams include integrals that require technique identification before computation. AB items test u-substitution, basic rules, long division, and completing the square. BC items add integration by parts, partial fractions, and improper integrals. A common task pattern presents an integral without labeling the method, requiring you to recognize the structure and apply the correct procedure accurately.
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open calculatorAP Calc Unit 6 covers 14 topics built around integration and accumulation of change. Key topics include Riemann Sums, the Fundamental Theorem of Calculus (Parts 1 and 2), accumulation functions, definite and indefinite integrals, u-substitution, and integration using long division. BC students also cover integration by parts, partial fractions, and improper integrals. Here's the full topic list: - 6.1 Exploring Accumulations of Change - 6.2 Approximating Areas with Riemann Sums - 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation - 6.4 The Fundamental Theorem of Calculus and Accumulation Functions - 6.5 Interpreting the Behavior of Accumulation Functions Involving Area - 6.6 Applying Properties of Definite Integrals - 6.7 The Fundamental Theorem of Calculus and Definite Integrals - 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation - 6.9 Integrating Using Substitution - 6.10 Integrating Functions Using Long Division and Completing the Square - 6.11 Integration by Parts (BC only) - 6.12 Linear Partial Fractions (BC only) - 6.13 Evaluating Improper Integrals (BC only) - 6.14 Selecting Techniques for Antidifferentiation See AP Calc Unit 6 for matched practice on every topic.
Unit 6 makes up 17-20% of the AP Calculus exam, making it one of the heaviest-weighted units on the test. It covers integration and accumulation of change, including Riemann Sums, the Fundamental Theorem of Calculus, antiderivatives, u-substitution, and several advanced techniques for BC students. That weight means roughly 1 in 5 exam points connects to this unit, so it's worth serious attention.
The AP Calc Unit 6 progress check in AP Classroom includes both MCQ and FRQ parts drawn from the unit's 14 topics. The MCQ section tests skills like setting up Riemann Sums, applying properties of definite integrals, and evaluating antiderivatives using substitution or basic rules. The FRQ part typically asks you to interpret accumulation functions, apply the Fundamental Theorem of Calculus, and select appropriate antidifferentiation techniques. BC students also see questions on integration by parts, partial fractions, and improper integrals in their progress check. Practicing these topics before the progress check at AP Calc Unit 6 will help you spot which techniques you still need to sharpen.
AP Calc Unit 6 FRQs most often pull from the Fundamental Theorem of Calculus, accumulation functions, and selecting antidifferentiation techniques, so those are the topics to prioritize. A typical FRQ asks you to evaluate a definite integral, interpret what an accumulation function represents in context, or justify behavior using area under a curve. To practice, work through released College Board FRQs that involve integration, write out every step of your reasoning (not just the answer), and check that your notation for definite and indefinite integrals is clean. BC students should also practice integration by parts and improper integrals in FRQ format. Find topic-aligned practice at AP Calc Unit 6.
The best place to find AP Calc Unit 6 practice questions, including multiple-choice and practice test sets, is AP Calc Unit 6. That page has resources organized by topic, so you can target Riemann Sums, the Fundamental Theorem of Calculus, u-substitution, or any of the other 14 topics in this unit. For MCQ practice, focus on questions that ask you to evaluate definite integrals, interpret accumulation functions, or choose the right antidifferentiation technique. Released College Board exams are also a strong source for realistic practice test questions on integration.
Start AP Calc Unit 6 by building a solid understanding of Riemann Sums and definite integral notation before moving to the Fundamental Theorem of Calculus, since later topics stack on those foundations. Then work through antiderivative rules, u-substitution, and long division in order, checking your understanding with practice problems after each topic. A concrete study plan: review one topic per session, do at least five practice problems per topic, and then take a timed MCQ set at the end of the unit to see which techniques still feel shaky. BC students should budget extra time for integration by parts, partial fractions, and improper integrals. Keep your notation tight throughout. Definite integrals with wrong bounds or missing dx are common point losses on the exam. Use AP Calc Unit 6 to find topic-specific practice as you go.